Linear Pencils 1
Spectrums of Linear Pencils in Large Random Haar Unitaries
Spectral Radii of Large Random Unitary Pencils
Linear Pencil Spectral Radius Visualization — User Guide
This web application allows you to explore the spectral properties of random linear pencils using the Wolfram Cloud. It computes and visualizes the behavior of the spectral radius (the largest absolute value of eigenvalues) for a randomly generated tuple of matrices.
What you can do
- Size of the Coefficient Matrix (k): Choose 2, 3, or 4. Larger matrices increase computation time.
- Plot Style (2D or 3D): 2D shows eigenvalues in the complex plane; 3D adds an extra axis for enhanced visualization.
- Sample Size: Number of random matrices averaged when computing spectral radius trends.
- Matrix Start / Stop Size: Range of matrix sizes over which the spectral radius is analyzed.
- Fixed Matrix Size (d): Computes the spectral radius distribution at this size and shows it as a histogram.
- Number of Realizations: Number of independent random matrices generated at size d to estimate the spectral radius distribution.
What the app does when you click "Run"
- Generates a tuple of random coefficient matrices of the chosen size.
- Generates random Haar unitary matrices of appropriate dimensions.
- Computes and displays four plots:
- Spectral Radius Growth Plot — shows how average spectral radius grows with matrix size.
- Spectral Radius Variance Plot — shows variance across random samples.
- Spectral Radius Histogram — probability distribution at the chosen matrix size d.
- Spectrum of a Single Realization — visualizes eigenvalues for one random tuple.
- Combines all plots into a single grid image displayed in the app.
How to use
- Select your desired parameters using the form controls.
- Click Run.
- Wait for the app to finish computing. Computation time increases with matrix size and number of realizations.
- The resulting image will appear in the iframe below the form.
Notes
- The computations are randomized; each run may produce slightly different results.
- Larger values for k, d, or numRealization increase computation time.
- The app is publicly accessible — no login is required.
- The plots provide insight into the statistical behavior of random linear pencils for research, teaching, or exploratory experimentation.